CC BY
33Electronic Journal «Technical Acoustics» http://www.ejta.org
2007, 11
Patrick J. Vitarius1, Don A. Gregory2, Valentin Korman3, John Wiley4
1 FreelInnovations, Inc., Huntsville, AL, 35802, USA
2 The University of Alabama in Huntsville, Huntsville, AL 35899, USA
3 Madison Research Corporation, Marshall Space Flight Center, AL 35812, USA
4 Marshall Space Flight Center, Marshall Space Flight Center, AL 35812, USA
Correcting distortion in acoustic sense lines
Received 05.04.2007, published25.06.2007
Pressure sense lines, as employed in the measurement of rocket engine test firings, can propagate the time-domain pressure signal out of hostile regions, which allows the use of instrumentation with fragile pressure transducers. In such applications, it is necessary to correct the data to account for attenuation and resonance due to the sense line. One technique for doing this involves the application of Fourier transform theory to obtain the transfer function of the sense line. Various techniques for obtaining the transfer function are explored, including the use of Gaussian noise, single frequency sweeps, and impulse signals as input functions. The transfer function thus obtained is then mathematically fit, scaled, and validated against a related rocket engine test.
INTRODUCTION
The non-directional nature of acoustic waves makes a spatial array of sensors desirable when recording an acoustic event. Time delays between multiple sensors, as well as changes in amplitude and phase, can provide a more complete understanding of the acoustic event than if only a single sensor is used. The goal of achieving a symmetric array of well-placed sensors is complicated in propulsion measurements by the fact that the acoustic event is accompanied by a chemical combustion event that creates a hostile environment: high temperatures and pressures, toxic or oxidizing agents, and highly energetic micro-particles. The most interesting regions in a propulsion system test — those regions close to the propulsion system itself, and directly along its principal axis — may be the regions most difficult to instrument.
One approach for making the required acoustic measurements requires robust sensors, which may either be actively or passively cooled, and engineered to withstand a range of forces, whether due to rapid pressure fluctuations or the impingement of energetic particles. This paper considers an alternative approach that may prove, in many scenarios, more economically practical and easier to implement. This approach uses expendable sense lines made from Tygon® (or other) tubing that transmit the acoustic information from the hostile region to a remotely located sensor [1].
The use of sense lines is routine for static pressure measurements. A sense line that is exposed to a certain pressure at one end while closed on the other end will quickly reach that
pressure along its entire length. For dynamic pressure or acoustic measurements, resonance and attenuation cause distortion of the time-domain signal.
Irwin, Cooper, and Girard [2] describe a number of methods for approximating the original time-domain signal at the open end of the sense line given the time-domain signal at the closed end. These methods are derived from pressure data obtained from architectural model wind tunnel tests, but are presented in a general model. These methods have also been applied to another field, the analysis of airfoils [3].
This paper focuses on one method for correcting distorted sense line data, namely, correction by a multiplicative transfer function. The technique, as examined here, is an amplitude-only technique in that phase effects of the sense line are not considered. Various techniques for acquiring the transfer function are considered, and the experimentally determined transfer function is then re-scaled and validated with data from an actual rocket motor test firing.
The pressure as a function of time at the open end of a sense line is mathematically represented as the input signal, and the pressure at the transducer at the far end of the sense line is the corresponding output signal. The system is therefore the sense line itself, and encompasses any mode of propagation that links the two ends. Let the input and output signals be given by gt and go, and let the system operator be designated as S :
The input function can be rewritten by applying the sifting property of Dirac delta functions:
which is independent of the input function gi (t). This function h(t ,t) is called the impulse response function. With h thus defined, Equation (3) can be written
For systems or components whose characteristics do not change implicitly with time, h(t,T) is in fact only a function of the difference between t and t :
THEORY
go (t) = S{gi (t)}.
(1)
(2)
The operator S only acts on functions of t, and the integral is over all values of t , so the order of these two can be reversed:
go (t) = j gt (t)S(^(t -т)dт^,
(3)
and the operator acting on a delta function input can be defined as h(t, t) = S{S(t -t)},
(4)
go (t) = j gi (T)h(t,T)dT
(5)
h(t, t) = h(t - t)
(6)
so
go (t) = j gt (T)h(t -T)dT.
(7)
This is recognized to be the convolution of gi(t) and h(t) . From the convolution
theorem, the Fourier transform of a convolution is equal to the product of the Fourier transforms of its constituents. Thus
Go (a) = H (a)G,(a) (8)
or
H (a) =
Go (a)
Gt (a)
(9)
where Go(a) and Gi(a) are the Fourier transforms of the output and input functions and H(a) is both the amplitude transfer function and the Fourier transform of the impulse response function of the system. The uncertainty in the transfer function may be obtained by applying Taylor’s theorem,
„2 J<t\„2_ Jct\
№'v) la*J * Vdvj
to equation 9, obtaining
(10)
=
aG +
Go
'Gr
(11)
Similarly, when the output function and the transfer function are treated as variables and the input function is expressed in terms of them,
< =
+
Go H2
' h •
(12)
2
TECHNIQUE
Linear transform theory provides a way to at least partially correct acoustic signals that have been transmitted through a sense line. Equation (9) is used twice: first, input and output signals are measured to determine the amplitude transfer function, and second, the calculated transfer function is used with a measured output signal in order to form an approximation of the input signal.
The uncertainty in the input signal calculated using this method is a function of the measured quantities and their respective uncertainties. Equation (12) shows that the uncertainties will be large in regions where the transfer function is small (highly attenuated frequencies); Equation (11) shows that the uncertainties will also be large in regions where the input signal is weak. Thus considerable thought should be given to choosing an appropriate input signal for the first part of the procedure. One criterion for choosing an appropriate input signal should be reasonably high signal strength across a wide band of
frequencies. This suggests broad-band noise generators, impulse signals, and frequency sweeps as candidate input signals.
An additional consideration involves the non-linearity of the sense line. It is indeed true that, for an ideal linear system, a single transfer function can be used to correct any output function, but even for a non-linear system, it is reasonable to expect that similar input signals would experience similar attenuation effects from the sense line. Thus the transfer function should be calculated using an input signal as similar as possible to the one that will ultimately be sensed.
APPARATUS
An array of PCB Piezotronics, Inc. Model 102A10 Dynamic Pressure Transducers was installed to instrument the static test firing of a rocket engine. Each transducer, including those not directly within the line of fire, was protected by a three-foot sense line constructed from 0.25-inch (internal diameter) Tygon® tubing. The uniform use of sense lines allowed for a verification (by analysis of time delays) that the primary mode of propagation was along the sense line, and not directly from the source to the transducer.
Prior to the test, a sample transducer was attached to a 12-inch long 0.25-inch Tygon® sense line. A similar transducer was placed next to the open end of the sense line. The array was used to measure pressure signals from a variety of sources, including a Gaussian noise sequence and single-frequency sweeps generated by a commercially available function generator attached to a mid-frequency range speaker, and impulsive events including a rolled newspaper striking the floor, a book dropped from a constant height, and the popping of uniformly inflated latex balloons. The fast Fourier transforms (FFT’s) of each of these events was evaluated in terms of broadness, smoothness, and repeatability of the spectra. It was determined that the balloon pop was the most suitable input signal.
RESULTS
The amplitude transfer function obtained from the average of fifty balloon pops is shown in Figure 1 below. The periodic peaks are at frequencies corresponding to the resonant frequencies of the open-ended sense line. The transfer function obtained from the Gaussian had similar characteristics, but was of considerably poorer quality.
The amplitude transfer function of Figure 1 has two dominant traits: a periodic array of sharp peaks, modulated by a decaying envelope. In order to easily scale this transfer function to various sense lines, a simple mathematical function was developed that encompassed these traits. A sum of cosines was chosen for the periodic array, and an exponential decay above a constant floor for the envelope. Parameterized for length, the function may be written
Hfit (f) = O(f) E (f), (13)
where
O(f) =
600
230 - 170cos
+ 67 cos
+ 37cos
f 4nf_ ^ f1
V f1 J - 21cos
-48cos
J
J
f 3nf '
k/j
(14)
and
E (f) = 5.50 + 11.50e
2.47 f
(15)
1
Fig. 1. Amplitude transfer function obtained from the average of fifty balloon pops
The parameter f1 represents the fundamental frequency of the open-ended sense line, and is related to sense line length by:
v
f = —, (16) 4L
where vs is the speed of sound in air at ambient conditions, and L is the length of the sense
line. It is important to note that the transducers used here had cavities much smaller than the total volume of the sense line; for a sufficiently large transducer cavity the sense line is effectively open at both ends and different resonant conditions apply.
The data obtained via the three-foot sense lines was corrected by scaling the transfer function given in Equation (13) and dividing it into the output signal, as suggested by Equation (9). The power spectra of the output signal, before and after correction, as well as the actual input signal, are shown below (Figure 2). The actual signal was obtained from a sensor, without a sense line, that was placed sufficiently far from the rocket engine that the environment was not a factor.
detected
corrected
actual
Octave frequency bands (Hz)
Fig. 2. Detected, corrected, and actual power spectra (standard octave bands) for the three-
foot sense line data set
The benefits and limitations of the approach given in this paper can be seen by comparing the power spectra of Figure 2. In terms of the power density and the band-integrated power, the corrected signal is a more accurate picture of the acoustic signal. However, the corrected signal is much too high in the 200 Hz octave band, in part because of the herringbone pattern in the actual power spectrum (Figure 3) that tends to selectively amplify or diminish the effect of the transfer function at certain frequencies.
Fig. 3. A portion of the actual power spectrum demonstrates the herringbone pattern — rapid oscillation in frequency space between relatively high and low power. This herringbone pattern tends to selectively amplify or diminish the effect of the transfer function at certain
frequencies
CONCLUSION
The acoustic sense line and the fast Fourier transform are two traditional and readily accessible tools for propulsion engineers. These have historically been applied to static pressure measurements and unattenuated signals. However, in situations where an unattenuated signal is not obtainable, and a static (mean) pressure is less important than the dynamic pressure signal, a combination of these two techniques can produce a result that satisfies the immediate needs of the experimenter without resorting to expensive equipment or unfamiliar techniques. The simple example presented here illustrates the general usefulness of the technique without resorting to computer generated simulations of data. This acoustic data is real and was taken from an actual rocket motor test firing. From the figures, it is apparent how much data is completely lost if no correction is attempted. The correction presented here is only one of many possible choices but it does illustrate that even simple corrections can restore some of the original signal.
SYMBOL DEFINITIONS
E - decaying portion of transfer function
f - frequency
gi - acoustic input function
go - acoustic output function
Gi - Fourier transform of input function
Go - Fourier transform of output function
H - amplitude transfer function
h - impulse response function
L - sense line length
O - oscillatory portion of transfer function
S - system operator t - time
t - temporal integrating variable vs - speed of sound in air <jx - uncertainty (standard deviation) for quantity x
REFERENCES
1. Vitarius P., Gregory D. A., Wiley J., and Korman V. Acoustic Wave Propagation in Pressure Sense Lines. 39th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit, AIAA-2003-5182, AIAA, Washington, DC, 2003.
2. Irwin H. P. A. H., Cooper K. R., and Girard R. Correction of Distortion Effects Caused by Tubing Systems in Measurements of Fluctuating Pressures. J. Ind. Aerodyn., 5 (1979), pp. 93-107.
3. Swalwell K. E., Sheridan J., and Melbourne W. E. Frequency Analysis of Surface Pressures on an Airfoil After Stall. 21st AIAA Applied Aerodynamics Conference, AIAA-2003-3416, AIAA, Washington, D.C., 2003.
